Hi I'm new to set theory. I need to prove that if $A \subseteq B$, then $A \cap B = A$. I would like to do this the formal way, without a Venn diagram. How should I proceed?
[Math] How to show if $A \subseteq B$, then $A \cap B = A$
elementary-set-theory
Best Answer
The most straightforward way is to show that $A\cap B\subseteq A$ and $A\subseteq A\cap B$. Each of these inclusions can be proved by what I call element-chasing: let $x$ be an arbitrary element of the lefthand side, and show that $x$ is necessarily an element of the righthand side.
For the first inclusion this is trivial: if $x\in A\cap B$, then by definition $x\in A$, and it follows immediately that $A\cap B\subseteq A$.
The second inclusion is almost as easy. Suppose that $x\in A$. Then by hypothesis $x\in B$ (since $A\subseteq B$), so $x\in A\cap B$. (In words, is an element of both $A$ and $B$ and therefore by definition an element of $A\cap B$.)